Calculating Probabilities using z-values under the Standard Normal Distribution

Using the standard normal distribution table, we can be able to calculate the that a normally distributed random variable Z, with mean equal to 0 and variance equal to 1, is less than or equal to z, i.e., P(Z ≤ z).

However, the table does this when we only have positive values of z. Simply put, if the examiner asks you to find the probability behind a given positive z-value, you’d have to look it up directly on the table.

P(Z ≤ z) = θ(z) when z is positive

Example: Using the z-score table

Using the data from our first example, suppose you were asked to calculate the probability that the return is less than $1.

Solution

First, you’d be required to calculate the z-value (2 in this case).

P(Z ≤ 2) can be read off directly from the table.

You just move down and locate the z-value that lies to the right of “2” i.e., 0.9772.

Note: The table above is incomplete.

Negative z-values

If we have a negative z-value and do not have access to the negative values from the table (as shown below), we still can calculate the corresponding probability by noting that:

$$ P(Z \le -z) = 1 – P(Z \le z) \text{ or} $$

$$ \theta(–z) = 1 – \theta(z) $$

This relationship is true when we consider the following facts:

The total area (probability) under the standard normal distribution is 1.